Channel norm-based ordering and whitened decoding for MIMO communication systems

ABSTRACT

A channel norm-based ordering and whitened decoding technique (lower complexity iterative decoder) for MIMO communication systems performs approximately the same level of performance as an iterative minimum mean squared error decoder.

RELATED APPLICATIONS

[0001] This application is related to provisional application serial No.60/404,860, docket number TI-35082PS, filed Aug. 21, 2002, by David J.Love, Srinath Hosur, Anuj Batra, Tarik Muharemovic and Eko N.Onggosanusi.

BACKGROUND OF THE INVENTION

[0002] 1. Field of the Invention

[0003] This invention relates generally to multiple-inputmultiple-output (MIMO) communication systems. More particularly, thisinvention relates to channel norm-based ordering and whitened decodingfor MIMO communication systems.

[0004] 2. Description of the Prior Art

[0005] Multiple-input multiple-output (MIMO) communication systemsprovide gain in capacity and quality compared to single-inputsingle-output (SISO) communication systems. While a SISO system employsone transmitter and one receiver to receive the transmitted signal, aMIMO system in general uses M_(t)>=1 transmitters and M_(r)>=1receivers. Thus the SISO system is a special case of a MIMO system, withM_(t)=M_(r)=1. Examples of MIMO systems include but are not limitedto: 1. A communication system employing multiple-antennas at thetransmitter and/or receiver; 2. A communication system employingorthogonal frequency division multiplexing (OFDM) or code divisionmultiplexing (CDMA); 3. A time/frequency division multiple accesssystem; 4. Any multiuser communication system; 5. Any combination of 1-4above.

[0006] Typically, the MIMO systems consist of a MIMO transmitter thatsends “multidimensional” symbol information. This multidimensionalsymbol could, but is not limited to, be represented by a vector (notethat a matrix symbol can always be represented as a vector symbol bystacking the columns/rows of the matrix into a vector). Themultidimensional symbol might represent one or more coded or uncodeddata symbols corresponding to SISO transmitters. The transmitted signalpropagates through the channel and is received and processed by a MIMOreceiver. Note that the receiver could obtain multiple received signalscorresponding to each transmitted symbol. The performance of thecommunication system hinges on the ability of the receiver to processand find reliable estimates of the transmitted symbol based on thereceived signals.

[0007] Definitions

[0008] As used herein, bolded capitol symbols, such as H, representmatrices.

[0009] As used herein, bolded lower-case symbols, such as s, representvectors.

[0010] As used herein, ^(T) denotes matrix transposition.

[0011] As used herein, * denotes the matrix conjugate transposeoperation.

[0012] As used herein, ^(−l) denotes the matrix inverse operation.

[0013] As used herein, if W is a matrix, W_(m) denotes the mth column ofW.

[0014] As used herein, if W is a matrix, (W^(T))_(m) denotes the mth rowof W.

[0015] As used herein, if v is a vector, ∥v∥₂, denotes the 2-norm of v.

[0016] As used herein, if Q(^(.)) represents the symbol slicingfunction, it will be assumed to slice both single symbols andmulti-dimensional symbol vectors.

[0017] As used herein, I_(M) represents the M by M identity matrix.

[0018] As used herein, 0 _(M×N) represents the M by N matrix of zeros.

[0019] As used herein, if A and B are sets, then A|B is the set of allelements in A that are not in B.

[0020] For MIMO systems such as, but not limited to, the ones discussedherein above, the received signal can be written, after front endreceive processing such as filtering, downconversion, AGC,synchronization etc., in the form $\begin{matrix}{y_{k} = {{\sum\limits_{n}^{\quad}{H_{n}s_{k - n}}} + v}} & (1)\end{matrix}$

[0021] where H_(n) is an M_(r) by M_(t) matrix of complex gains, s_(k)is the M_(t)-dimensional symbol vector transmitted at time k, and v is aM_(t)-dimensional vector of additive noise. In narrowband wirelesssystems where the symbol period is much larger than the RMS delay spreadas well as in OFDM systems where the inter-symbol interference isnegligible due to the insertion of a cyclic prefix and/or guardinterval, the channel from each transmit antenna to each receive antenna(per frequency bin in case of OFDM) is often modeled as a single-tapcomplex gain. In this case equation (1) simplifies to

y _(k) =Hs _(k) +v  (2)

[0022] where H is now an M_(r) by M_(t) matrix of complex numbers andHs_(k) is the matrix product of H and s_(k).

[0023] The receiver must estimate the symbol matrix S=[s₁ . . . s_(T)]in order to facilitate reliable communication. Examples, but by no meansthe only examples, of multidimensional symbols could be space-time codeswhere T>1 or spatial multiplexing systems with T=1 and independent SISOmodulation on each transmit antenna. In case of no or negligibleadditive noise, v, and an invertible H, the estimation problem wouldreduce to that of inverting H. The presence of non-negligible noise,however, increases the difficulty in estimating S. Note that we haveassumed that the receiver is has some estimate of H, that could beobtained by transmitting appropriate training sequences. The symbolmatrix S is also assumed to be chosen from a finite set C of possiblemultidimensional symbols (this is typically the case as for e.g. whereeach element of S is chosen from a QAM symbol set).

[0024] The optimal solution in the sense of minimizing the probabilityof symbol error has been shown to be the maximum aposterior (MAP)decoder which in case of equiprobable symbol transmissions is equivalentto a maximum likelihood (ML) decoder. The ML decoder attempts to find S,the symbol matrix, by using the symbol matrix {tilde over (S)} thatmaximizes p({tilde over (S)}|y₁, . . . , y_(T)) where p (·|y₁, . . . ,y_(kT)) is the conditional probability density function (pdf) of s_(k)given y₁, . . . , y_(T). In real-time communications systems, however,this type of decoder is overly computationally complex. Decoders thatsearch over a set V of possible multidimensional symbols S and decode tothe multidimensional symbol {tilde over (S)} in V that minimizes somesort of metric are denoted as minimum distance (MD) decoders. The MAPand ML decoders are MD decoders with V=C, where C is the set of allpossible multidimensional symbols S.

[0025] Many algorithms that are computationally easier than ML decodinghave been proposed in order to overcome the huge computational burden ofML decodings. Algorithms that perform some form of reduced complexitydecoding will be referred to herein as sub-optimal decoders. An exampleof a suboptimal decoder is successive interference cancellation (SIC)method. A receiver using SIC decodes each symbol within the symbolvector one at a time. After each symbol is decoded, its approximatecontribution to the received vector is subtracted in order to improvethe estimate of the next symbol within the symbol vector. The order ofsymbol decoding and subtraction could be arbitrary or based on rulessuch as maximization of pre/post processing SNR etc.

[0026] An example of an SIC receiver is the ordered iterative minimummean squared error (IMMSE) receiver. With a single-tap channel, thereceive signal is given by equation (2) above. Letting s_(k)=[s₁ s₂ . .. s_(Mr)]^(T), the ordered IMMSE operates using the following steps,letting y_(k,0)=y_(k), D₀={1,2, . . . , M_(t)}, and H_(k) ⁽⁰⁾=H.

[0027] 1. Set m=0.

[0028] 2. Compute W^(m)=(H_(k) ^((m)*)H_(k) ^((m))+pI_(Mt−m))⁻¹H_(k)^((m)*). $\begin{matrix}\begin{matrix}{{{3.\quad {Let}\quad n} = {\arg \quad {\min\limits_{i \in D_{0}}{{\left( W^{{(m)}T} \right)_{i}}_{2}.}}}}} \\{{{4.\quad {Set}\quad {{\overset{\sim}{s}}_{k,n} \cdot}} = {{Q\left( {y_{k,m}^{T}\left( W^{{(m)}T} \right)}_{i} \right)}.}}}\end{matrix} \\{{{{5.\quad {Set}\quad y_{k,{m + 1}}} = {y_{k,m} - {H_{k,n}^{(m)}{\overset{\sim}{s}}_{k,n}}}},{D_{m + 1} = {D_{m}\backslash \left\{ n \right\}}},\quad {and}}\quad}\end{matrix}$  H_(k)^((m + 1)) = [H_(k, 1)^((m + 1))H_(k, 2)^((m + 1))  …  H_(k, n − 1)^((m + 1))0_(M  r × 1)H_(k, n + 1)^((m + 1))  …  H_(k, M  t)^((m + 1))].

[0029] 6. Repeat steps 1-5 for m<M_(t).

[0030] 7. Set the decoded symbol vector to {tilde over (s)}_(k)=[{tildeover (s)}_(k,1) {tilde over (s)}_(k,2) . . . {tilde over(s)}_(k,M1)]^(T).

[0031] Regarding the above algorithm, it is important to note thatH_(k,i) ^((m+1)) denotes the ith row of the matrix H_(k) ^((m+1)) (timek and iteration m+1). Another example among sub-optimal decoders is thezero-forcing decoder which decodes to the symbol {tilde over(s)}_(k)=Q(H⁻¹y_(k)). This decoder is usually considered the worstperforming and least complex of the sub-optimal decoders.

[0032] Sub-optimal techniques unfortunately differ in diversity orderfrom ML decoding (i.e. the asymptotic slope of the average probabilityof bit error curve). They essentially trade reduced complexity forreduced performance.

[0033] In view of the foregoing, it is both advantageous and desirableto provide a lower complexity iterative decoder with channel norm basedordering that performs approximately the same as the iterative minimummean squared error (IMMSE) decoder.

SUMMARY OF THE INVENTION

[0034] The present invention is directed to a channel norm-basedordering and whitened decoding technique (lower complexity iterativedecoder) for MIMO communication systems and that performs approximatelythe same as the iterative minimum mean squared error decoder.

[0035] According to one embodiment, a method of decoding a signal vectorcomprises the steps of receiving a signal vector y_(k); multiplying thereceived signal vector y_(k) by a conjugate transpose of a channelmatrix H* and generating a column vector z_(k) therefrom; reorderingentries associated with the column vector z_(k) and generating anestimated channel matrix {tilde over (H)} therefrom; decomposing theestimated channel matrix {tilde over (H)} via Cholesky decomposition andgenerating a triangular matrix L therefrom; solving triangular matrix Lbackwards and estimating a signal vector {tilde over (s)}_(k) therefrom,wherein {tilde over (s)}_(k) is the true sorted symbol vector; andsorting signal vector {tilde over (s)}_(k) and generating an estimate ofthe transmitted symbol vector {tilde over (s)}_(k) therefrom.

BRIEF DESCRIPTION OF THE DRAWIMGS

[0036] Other aspects and features of the present invention and many ofthe attendant advantages of the present invention will be readilyappreciated as the same become better understood by reference to thefollowing detailed description when considered in connection with theaccompanying drawings wherein:

[0037]FIG. 1 is a system block diagram illustrating a decoder accordingto one embodiment of the present invention;

[0038]FIG. 2 is a graph depicting simulation results illustrating acomparison of the decoder shown in FIG. 1 with an ordered IMMSE for a2×2 system using 16-QAM; and

[0039]FIG. 3 is a graph depicting simulation results illustrating acomparison of the decoder shown in FIG. 1 with an ordered IMMSE for a2×2 system using 64-QAM.

[0040] While the above-identified drawing figures set forth particularembodiments, other embodiments of the present invention are alsocontemplated, as noted in the discussion. In all cases, this disclosurepresents illustrated embodiments of the present invention by way ofrepresentation and not limitation. Numerous other modifications andembodiments can be devised by those skilled in the art which fall withinthe scope and spirit of the principles of this invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0041]FIG. 1 is a system block diagram illustrating a decoding algorithm100 according to one embodiment of the present invention. The decodingalgorithm 100 was found by the present inventors to provide reducedcomplexity without much loss in performance when compared with knownsub-optimal schemes based on iterative interference cancellation. Inorder to preserve clarity and brevity, and to better understand howdecoding algorithm 100 functions, it will be assumed that each randomentry in v is complex Gaussian with variance σ² and that the expectedvalue of the transmit power on each antenna is constrained to be 1. Areceived symbol vector y_(k) is multiplied by the conjugate transpose ofthe channel matrix yielding

z _(k) =H*Hs _(k) +H*v.  (3)

[0042] The entries in the column vector z_(k) are then reordered basedon the column norms of the channel. This is done by defining

{tilde over (z)} _(k) =[z _(k,i) ₁ , z _(k,i) ₂ , . . . , z _(k,i) _(Mt)]^(T) , {tilde over (s)} _(k) =[s _(k,i) ₁ , s _(k,i) ₂ , . . . , s_(k,i) _(Mt) ]^(T)  (4)

and

{tilde over (v)} _(k) =[v _(k,i) ₁ , v _(k,i) ₂ , . . . , v _(k,i) _(Mt)]^(T), and {tilde over (H)}=[H _(k,i) ₁ , H _(k,i) ₂ , . . . , H _(k,i)_(Mt) ]  (5)

[0043] where for m≦n, ∥H_(k,k) _(m) ∥₂≦∥H_(k,i) _(n) ∥₂. At this point,a Cholesky decomposition is taken such that

L*L={tilde over (H)}*{tilde over (H)}+σ ² I _(M) ₁ .  (6)

[0044] Now multiplying {tilde over (z)}_(k) by L^(*−1) yields

x _(k) =L ^(*−1)({tilde over (H)}*{tilde over (H)}+σ ² I _(M) _(t)){tilde over (s)} _(k) −L ^(*−1)σ² I _(M) _(t) {tilde over (s)} _(k) +L^(*−1) {tilde over (H)}*{tilde over (v)}=L{tilde over (s)} _(k) +L^(*−1)({tilde over (H)}*{tilde over (v)}−σ² I _(M) _(t) {tilde over (s)}_(k))  (7)

[0045] where {tilde over (s)}_(k) is the true sorted symbol vector. Thetriangular matrix L is then used to solve backwards for {tilde over(s)}_(k). After estimating {tilde over (s)}_(k), the vector is thensorted into an estimate of the transmitted symbol vector ŝ_(k). Thedecoding algorithm 100 returns this estimated vector along with the softand hard decoded bits as shown in FIG. 1.

[0046] It is important to note that the noise+interference term duringthe backwards substitution step, L^(*−1)({tilde over (H)}*v−σ²I_(M) _(t){tilde over (s)}_(k)), is zero mean with a covariance matrix of σ²I_(M)_(t) under the assumption that the transmitters transmit each point inthe constellation with equal probability. This is important if an outercode such as a convolutional code is used since it allows the Viterbialgorithm to be used as an optimal ML decoding scheme for the outercode. The IMMSE algorithm does not have this property.

[0047] The computational complexity of the decoding algorithm 100 shownin FIG. 1 is also lower than ordered IMMSE. This algorithm 100 requiresordering only once during the decoding. In contrast, the ordered IMMSEalgorithm requires M, orderings while decoding a symbol vector. Thealgorithm 100 shown in FIG. 1 and discussed herein above, can be seen toalso require a Cholesky decomposition and the inversion of a triangularmatrix. In contrast, the ordered IMMSE algorithm requires M,−1 matrixinversions.

[0048] The algorithm 100 was simulated for a system with 2 inputs and 2outputs experiencing MIMO Rayleigh fading using quadrature amplitudemodulation (QAM) for various constellation sizes. In MIMO Rayleighfading, the channel matrix H is a random matrix where all the entries ofH are independent, identically distributed complex Gaussian randomvariables. Simulations shown in FIG. 2 and FIG. 3 show that the raw (oruncoded) average probability of bit error of the algorithm 100 comparesfavorably with ordered IMMSE decoding. In fact, for 16-QAM 200,algorithm 100 can be seen to outperform ordered IMMSE decoding.

[0049] In summary explanation of the above, a column norm-based ordered,whitened decoding algorithm 100 is provided for MIMO communicationssystems. The decoding algorithm 100 has been shown to performapproximately the same as ordered iterative minimum mean squared errordecoding while incurring less computational complexity. The decodingalgorithm 100 as well does not color the noise between symbol decisionswhen decoding a symbol vector.

[0050] In view of the above, it can be seen the present inventionpresents a significant advancement in the art of multiple-inputmultiple-output (MIMO) communication systems. Further, this inventionhas been described in considerable detail in order to provide thoseskilled in the decoding art with the information needed to apply thenovel principles and to construct and use such specialized components asare required.

[0051] Further, it should be apparent that the present inventionrepresents a significant departure from the prior art in constructionand operation. However, while particular embodiments of the presentinvention have been described herein in detail, it is to be understoodthat various alterations, modifications and substitutions can be madetherein without departing in any way from the spirit and scope of thepresent invention, as defined in the claims which follow. For example,the although the inventive embodiments described herein before weredescribed in association with a 2×2 system, the invention is not solimited, and can just as easily be applied to any Mt×Mr system.

What is claimed is:
 1. A method of decoding a signal vector, the methodcomprising the steps of: receiving a signal vector y_(k); multiplyingthe received signal vector y_(k) by a conjugate transpose of a channelmatrix H* and generating a column vector z_(k) therefrom; reorderingentries associated with the column vector z_(k) and generating anestimated channel matrix {tilde over (H)} therefrom; decomposing theestimated channel matrix {tilde over (H)} via Cholesky decomposition andgenerating a triangular matrix L therefrom; solving triangular matrix Lbackwards and estimating a signal vector {tilde over (s)}_(k) therefrom,wherein {tilde over (s)}_(k) is the true sorted symbol vector; andsorting signal vector {tilde over (s)}_(k) and generating an estimate ofthe transmitted symbol vector {tilde over (s)}_(k) therefrom.
 2. Themethod according to claim 1, wherein the received signal vector y_(k) isrepresented by the relationship y_(k)=Hs_(k)+v and the column vectorz_(k) is represented by the relationship z_(k)=H*Hs_(k)+H*v, wherein His a matrix of complex numbers, s_(k) is a multidimensional symbolvector transmitted at time k, v is a multidimensional vector of additivenoise+interference, and Hs_(k) is the matrix product of H and s.
 3. Themethod according to claim 2 wherein the multidimensional vector ofadditive noise+interference v, is represented by the relationshipL^(*−1)({tilde over (H)}*v−σ²I_(M) _(t) {tilde over (s)}_(k)), andfurther wherein v has a zero mean value with a covariance matrix definedas σ²I_(M) _(t) under the assumption that associated communicationsystem transmitters transmit each point in the associated communicationsystem constellation with equal probability.